Non-dispersive wave packets in periodically driven quantum systems

454 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547
The quantum quasi-energy levels are then given by Eq. (102) which reads
E ;k=0 = − 3
2n2 −
0
3
8n4 a ( ; q) : (149)
0
As discussed in Section 3.1.2, the non-dispersive wave packet with maximum localization is associated
with = 0 and is well localized inside the non-linear resonance island between the internal
Coulombic motion and the external driving, provided the parameter q is of the order of unity (below
this value, the resonance island is too small to support a localized state). The minimum scaled
microwave amplitude is thus of the order of
F0;trapping = Ftrappingn 4 0 1
n2 ; (150)
0
which is thus much smaller—by a factor n2 0 , i.e., three orders of magnitude in typical experiments—
than the electric eld created by the nucleus. This illustrates that a well chosen weak perturbation
may strongly in uence the dynamics of a non-linear system. From the experimental point of view,
this is good news, a limited microwave power is su cient to create non-dispersive wave packets.
We have so far given a complete description of the dynamics of the resonantly driven, onedimensional
Rydberg electron, from a semiclassical as well as from a quantum mechanical point
of view, in the resonant approximation. These approximate treatments are now complemented by a
numerical solution of the exact quantum mechanical eigenvalue problem described by the Floquet
equation (75), with H0 and V from Eqs. (113) and (114), as well as by the numerical integration of
the classical equations of motion derived from Eq. (137). Using this machinery, we illustrate some
of the essential properties of non-dispersive wave packets associated with the principal resonance in
this system, whereas we postpone the discussion of other primary resonances to Section 5.3.
Fig. 10 compares the phase space structure of the exact classical dynamics generated by the
Hamilton function (137), and the isovalue curves of the pendulum dynamics, Eq. (144), for the
case n0 = 60, at scaled eld strength F0 = Fn4 0 =0:01. The Poincare surface of section is taken
at phases !t = 0 (mod 2 ) and plotted in (I; ) variables which, for such times, coincide with the
(Î; ˆ ) variables, see Eqs. (60) and (61). Clearly, the pendulum approximation predicts the structure
of the invariant curves very well, with the resonance island surrounding the stable periodic orbit
at (Î ≈ 60; ˆ = ), the unstable xed point at (Î ≈ 60; ˆ = 0), the separatrix, and the rotational
motion outside the resonance island. Apparently, only tiny regions of stochastic motion invade the
classical phase space, which hardly a ects the quality of the pendulum approximation. It should be
emphasized that—because of the scaling laws, see Section 3.2.5—the gure depends on the scaled
eld strength F0 only. Choosing a di erent microwave frequency with the same scaled eld leads,
via Eq. (139), to a change of n0 and, hence, of the scale of I.
Fig. 11 compares the prediction of the Mathieu approach, Eq. (149), for the quasi-energy levels
of the Floquet Hamiltonian to the exact numerical result obtained by diagonalization of the full
Hamiltonian, see Section 3.2. Because of the ˝! periodicity of the Floquet spectrum, the sets of
energy levels of the pendulum, see Fig. 9, are folded in one single Floquet zone. For states located
inside or in the vicinity of the resonance island—the only ones plotted in Fig. 9a—the agreement
is very good for low and moderate eld strengths. Stronger electromagnetic elds lead to deviations
between the Mathieu and the exact result. This indicates higher-order corrections to the pendulum
approximation.